Logarithmic Conformal Field Theory and Seiberg-Witten Models

TL;DR

This paper establishes a correspondence between periods of abelian forms on hyperelliptic Riemann surfaces, especially the Seiberg-Witten differential, and conformal blocks in a specific logarithmic conformal field theory, linking complex geometry with CFT.

Contribution

It reveals a novel connection between hyperelliptic curve periods and rational logarithmic CFT with central charge -2, providing explicit expressions using Lauricella hypergeometric functions.

Findings

Periods correspond to conformal blocks in the c=-2 logarithmic CFT.

Generic periods can be expressed via Lauricella hypergeometric functions.

The theory models the branched double covering of hyperelliptic curves.

Abstract

The periods of arbitrary abelian forms on hyperelliptic Riemann surfaces, in particular the periods of the meromorphic Seiberg-Witten differential, are shown to be in one-to-one correspondence with the conformal blocks of correlation functions of the rational logarithmic conformal field theory with central charge c=c(2,1)=-2. The fields of this theory precisely simulate the branched double covering picture of a hyperelliptic curve, such that generic periods can be expressed in terms of certain generalised hypergeometric functions, namely the Lauricella functions of type F_D.